1. The Lawful and Unlawful In regards Parents and Children
Paternity and Adoption

2. The child is an extension of his father and the bearer of his characteristics.
Allah has ordained marriage and has forbidden adultery so that paternity may be established without doubt or ambiguity and that the child may be referred to his father and the father to his sons and daughters.
Through marriage a woman is reserved for one man; it is haram for her to be unfaithful to him or to let anyone else have access to what belongs exclusively to him. Thus, every child born to her in wedlock will be her husband's child, without any need for recognition or public proclamation of the fact by him or a corresponding claim on the part of the mother.
"The child is attributed to the one on whose bed it is born,"
(Bukhari and Muslim)

3. It is not permissible for the husband to deny his paternity of any child born to his wife as long as they are married to each other. Such a denial would bring the ugliest shame imaginable upon both the wife and the child.
If however a man suspects his wife of not being faithful and the child she carries to not be his, the laws of islam give him a means of dealing with this as follows:
He can take the case to a Muslim judge(qadi). The qadi will ask the man and his wife to invoke the curse of Allah on one another in the manner prescribed in Surah al-Nur:
“As for those who accuse their wives but have no witnesses except themselves, the testimony of one of them shall consist of bearing witness by Allah four times that he is of the truthful, and a fifth (time) that the curse of Allah be upon him if he is one of those who lie. And it shall avert the punishment from her if she bear witness by Allah four times that he is indeed of those who lie, and a fifth (time) that the wrath of Allah be upon her if he is among the truthful.” (24:6-9)
After this the two shall be separated permanently, and the child shall be identified by the name of his or her mother.

4. Just as it is haram for a man to deny his paternity of a child born to his wife in wedlock, it is likewise haram for him to legally adopt a child of whom he is not the natural father
Like peoples of other societies during the course of history, the Arabs used to add anyone they wished to their lineage and family through adoption.
Before receiving the call to prophethood, the Prophet had himself adopted Zaid bin Harithah.Hakim bin Hizam had bought him for his aunt Khadijah, and after her marriage to the Prophet Khadijah presented Zaid to him. When Zaid's father and uncle learned his place of residence, they came to the Prophet to demand Zaid's return. The Prophet gave Zaid a choice, and he chose to stay with the Prophet in preference to his father and uncle. The Prophet then set him free and adopted him as his son in the presence of others.
He was thereafter called Zaid ibn Muhammad and became the first of the freed slaves to accept Islam.

5. Later Allah outlawed adoption making it unlawful for the following reasons:
Taking a stranger into the family as one of its members; and allowing him privacy with women who are not his muharrmat, nor he theirs, is a deception, for the man's wife is not the adopted son's mother, nor is his daughter the boy's sister nor is his sister his aunt, since all of them are non-mahrem to him.
Moreover, the adopted son acquires a claim on the inheritance of the man and his wife, depriving the rightful, deserving relatives of their inheritance.
Such a situation arouses the anger of the real relatives against the intruder who encroaches upon them and usurps their rights, depriving them of their full inheritance.

6. So thus Allah abolished adoption:
“...Nor has He made your adopted sons your (real) sons; that is simply a saying of your mouths. But Allah speaks the truth, and He guides you to the (right way). Call them by (the names of) their fathers; that is more just in the sight of Allah. But if you do not know their fathers, they are your brothers-in-faith and your wards....” (33:4- 5)

7. By outlawing adoption islam abolished all the effects of this system which relate to inheritance and marriage:
In matters of inheritance, the Qur'an does not recognize any claim except those based on relationship through blood and marriage:
“...But blood relatives are nearer to each other in the ordinance of Allah...”.
(8:75)
With regard to marriage, The Qur'an declared that only the wives of one's real sons,
"the wives of your sons who are from your (own) loins" (4:23),
not the wives of the adopted sons, are permanently forbidden in marriage.
Accordingly, it is permissible for a man to marry the divorced wife of his adopted son, since she has been, in actuality, the wife of a "stranger" not related by blood.

8. After Allah sent down the verses abolishing adoption he no longer referred to Zaid as his son and married Zaid's ex wife Zainab:
“And when thou didst say to him who had received the favor of Allah and thy favor, 'Retain thy wife and fear Allah,' thou didst hide within thyself what Allah was about to make manifest, fearing the people; but Allah has more right that thou shouldst fear Him. Then, when Zaid had carried out the necessary formality (of divorce) from her, We gave her to thee in marriage so that (in the future) there might be no difficulty for the Believers with respect to (marriage to) the wives of their adopted sons when the latter have carried out the necessary formality (of divorce) from them; and Allah's command must be fulfilled. “(33:37)